1.9 Standard form

Learning objectives

1.9A Standard formcalculate with and interpret numbers in the form a × 10ⁿ, where n is an integer and 1 ≤ a < 10 1.9HA Problems involving standard formsolve problems involving standard form

Standard form writes a non-zero number as $a imes10^n$ , where $1\le |a|<10$ and $n$ is an integer. The power of 10 records the place-value shift.

Ordinary number	Standard form
71 800 000	$7.18 imes10^7$
0.00042	$4.2 imes10^{-4}$
$-630000$	$-6.3 imes10^5$

For multiplication, multiply coefficients and add powers; for division, divide coefficients and subtract powers. For addition or subtraction, first express terms with the same power of 10.

Calculation	Normalised result
$(3 imes10^4)(2 imes10^5)$	$6 imes10^9$
$(8 imes10^7)/(4 imes10^3)$	$2 imes10^4$
$4.5 imes10^6+7 imes10^5$	$4.5 imes10^6+0.7 imes10^6=5.2 imes10^6$

A result such as $18 imes10^5$ is not in standard form because the coefficient is not below 10; rewrite it as $1.8 imes10^6$ .

Translate each contextual quantity into a consistent unit, preserve its standard-form structure through the calculation, then interpret and round the final result as requested.

Step	Healthcare spending per person
form a rate	total spending $\div$ population
Austria	$(4.2 imes10^{10})/(8.7 imes10^6)=4.82758\ldots imes10^3$
Luxembourg	$(3.7 imes10^9)/(6.3 imes10^5)=5.87301\ldots imes10^3$
compare	$5873.01\ldots-4827.58\ldots=1045$ dollars nearest whole

Separate coefficient arithmetic from exponent arithmetic, but keep enough calculator precision until the final requested rounding.

Use exponent size to check order of magnitude. Dividing $10^{10}$ by $10^6$ should produce a value around $10^4$ , subject to the coefficient ratio.

Do not subtract exponents when subtracting numbers. Exponent subtraction belongs to division; numerical subtraction requires compatible place values.

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—1.10 Applying numberRead

1.9 Standard form

Learning objectives

Write, interpret and calculate in standard form

Solve problems using standard form

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